An agent-based approach to procedural city generation incorporating Land Use and Transport Interaction models

Abstract: We apply the knowledge of urban settings established with the study of Land Use and Transport Interaction (LUTI) models to develop reward functions for an agent-based system capable of planning realistic artificial cities. The system aims to replicate in the micro scale the main components of real settlements, such as zoning and accessibility in a road network. Moreover, we propose a novel representation for the agent's environment that efficiently combines the road graph with a discrete model for the land. Our system starts from an empty map consisting only of the road network graph, and the agent incrementally expands it by building new sites while distinguishing land uses between residential, commercial, industrial, and recreational.

• The paper introduces an agent-based reinforcement learning approach using Deep Q-Learning to procedurally generate urban land use maps optimized by a reward function incorporating Land Use and Transport Interaction (LUTI) principles.
• A key technical aspect is integrating grid-based land use with graph-based road networks to accurately calculate accessibility, which is crucial for the agent's decisions and the LUTI-derived utility scores.
• The framework simulates realistic urban phenomena like zoning and accessibility dynamics through the agent's learned policy, balancing local land use compatibility with transportation network influence.

Agent-Based Procedural City Generation with LUTI Integration

The paper "An agent-based approach to procedural city generation incorporating Land Use and Transport Interaction models" (2211.01959) presents a methodology for generating artificial city layouts by assigning land uses to discrete plots within a predefined road network. The approach utilizes a single reinforcement learning agent, trained using Deep Q-Learning (DQN), to sequentially select land use types (residential, commercial, industrial, recreational) for undeveloped cells. A core aspect of this work is the incorporation of principles derived from Land Use and Transport Interaction (LUTI) models directly into the agent's reward function, aiming to produce more realistic urban structures compared to purely geometric or rule-based procedural generation methods.

Agent Framework and LUTI-Based Reward Structure

The system operates on a grid representing potential land plots, overlaid with an immutable road network imported from sources like OpenStreetMap or SUMO. A single "builder" agent selects undeveloped grid cells, prioritized by their accessibility, and assigns one of the four available land use types. The agent's state is defined by a tensor capturing the land use types and normalized accessibility values of cells within a local perception radius K.

The integration of LUTI principles is primarily achieved through the reward mechanism. The global reward, maximized by the DQN agent, is the sum of individual scores for all developed cells. The score for a cell depends on its assigned land use and its context, defined by LUTI-inspired formulae:

• Residential Score ($s_h$): Positively influenced by the cell's accessibility ($a_c$) and the number of nearby residential ($N_h$), commercial ($N_c$), and recreational ($N_r$) cells, while being negatively impacted by nearby industrial cells ($N_i$) within radius K.

  $$s_h = w_a a_c + w_{hh} N_h + w_{hc} N_c - w_{hi} N_i + w_{hr} N_r$$

• Commercial Score ($s_c$): Positively influenced by accessibility ($a_c$) and nearby residential cells ($N_h$). It exhibits a non-linear relationship with nearby commercial cells ($N_c$) to model market saturation effects (initially positive, then potentially negative) and is negatively influenced by nearby industrial cells ($N_i$).

  $$s_c = w_a a_c + w_{ch} N_h + w_{cc} N_c - w_{cc2} N_c^2 - w_{ci} N_i$$

• Industrial Score ($s_i$): Favors peripheral locations, using the distance to the nearest network node ($d_N(c)$) rather than overall accessibility. It is negatively influenced by nearby residential ($N_h$), commercial ($N_c$), and recreational ($N_r$) cells.

  $$s_i = w_{id} d_N(c) - w_{ih} N_h - w_{ic} N_c - w_{ir} N_r$$

• Recreational Score ($s_r$): Primarily influenced positively by nearby residential cells ($N_h$).

  $s_r = w_{rh} N_h$

The weights ($w$) in these functions are hyperparameters that encode the relative importance of accessibility and inter-land use adjacencies, effectively defining the zoning rules and utility perceptions the agent learns to optimize. The agent learns a policy $\pi(s)$ via DQN with experience replay to select the action (land use type) that maximizes the expected cumulative reward, leading to a final land use map.

Integrated Representation of Road Network and Land Use Grid

A significant contribution is the method used to integrate the graph-based road network and the grid-based land use representation for calculating accessibility. The road network is a directed graph $G = (V, E)$ where edge weights represent travel times. The land use map is a grid $C$.

1. Cell-to-Edge Mapping: Each grid cell $c \in C$ is associated with its geometrically nearest edge $e^c \in E$. Let $p_c$ be the point on $e^c$ closest to the center of cell $c$.
2. Inter-Cell Travel Time: The travel time $t(c_i, c_j)$ between two cells $c_i$ and $c_j$ is computed by summing the travel time from $c_i$ to $p_{c_i}$ (Euclidean distance scaled by a speed factor), the time along edge $e^{c_i}$ from $p_{c_i}$ to the edge's target node $e^{c_i}_t$, the shortest path time on the graph $G$ between $e^{c_i}_t$ and the source node of $e^{c_j}$, $e^{c_j}_s$ (using Dijkstra's algorithm), the time along edge $e^{c_j}$ from $e^{c_j}_s$ to $p_{c_j}$, and finally the time from $p_{c_j}$ to $c_j$.
3. Accessibility Metric: The accessibility $T(c)$ of a cell $c$ is the inverse of the average travel time to all other cells $c_i \in C$: $$T(c) = \left( \frac{1}{|C|} \sum_{c_i \in C} t(c, c_i) \right)^{-1}$$. This value is normalized to $[0, 1]$ to get $a_c$ used in the reward functions.
4. Computational Optimization: Calculating all-pairs shortest paths for $T(c)$ is computationally expensive ($O(|C|^2 \times \text{Dijkstra})$). The paper proposes partitioning cells based on their nearest edge ($C_e = \{c \in C | e^c = e\}$). By pre-calculating sums of travel times within these partitions ($T_e = \sum_{c_i \in C_e} t(c, c_i)$) and utilizing shortest paths between edge nodes, the calculation is accelerated, although the fundamental complexity remains quadratic in the number of cells. This optimization makes the approach feasible for moderately sized maps (tested up to 4096 cells or 16 km²).

Simulation of Zoning and Accessibility Dynamics

The framework simulates key urban phenomena through the agent's optimization process:

• Zoning: Explicit zoning rules are encoded in the reward function's interaction terms (e.g., attraction between residential and commercial/recreational, repulsion between residential and industrial). The agent learns to arrange land uses spatially to maximize aggregate scores based on these local neighborhood interactions, leading to emergent zoning patterns.
• Accessibility: Accessibility, derived from the road network topology and travel times via the integrated representation, plays a crucial role. It directly influences the utility scores for residential and commercial uses, promoting their concentration in well-connected areas. It also dictates the order of development, with the agent prioritizing more accessible cells, mimicking a common pattern in urban growth. Industrial placement is inversely related to centrality via the $d_N(c)$ term.

The resulting city layouts reflect a balance between maximizing local land use compatibility (zoning) and leveraging the transportation network (accessibility), consistent with principles observed in LUTI models. The agent acts as a centralized planner optimizing a specific utility function defined by the reward structure.

Methodological Considerations and Limitations

Strengths:

• Grounds procedural generation in established urban modeling concepts (LUTI).
• Provides a functional mechanism for integrating distinct spatial representations (graph network, grid land use) for accessibility calculation.
• Explicitly models accessibility and land use interactions via the reward function.
• Leverages reinforcement learning to potentially find non-trivial optimal land use allocations.
• Can utilize real-world road network data, enhancing the plausibility of the generated context.

Limitations:

• Scalability: The accessibility calculation remains a bottleneck, limiting practical application to moderate city sizes due to its $O(|C|^2)$ complexity, even with optimizations.
• Static Network: The road network is fixed and predefined; the system does not model the co-evolution of transportation infrastructure and land use.
• Simplified LUTI: The reward functions are specific mathematical forms representing a simplification of complex socio-economic interactions and utility preferences. Parameter tuning ($w$) is crucial and potentially subjective.
• Single-Agent Paradigm: Real urban development results from decentralized decisions by numerous heterogeneous agents (households, firms, developers). This model uses a single, centralized agent optimizing a global function.
• Evaluation: Assessment relies heavily on the internal reward score and visual inspection, lacking rigorous comparison against quantitative metrics of real-world urban form or alternative generative models.
• Path Dependency: The final configuration is highly sensitive to the initial road network structure and the sequential nature of the agent's decisions.

Potential Applications

Despite limitations, the approach offers potential in several areas:

• Urban Planning Support: As a rapid prototyping tool for exploring land use scenarios on given networks and evaluating them based on LUTI-derived metrics.
• Virtual Environment Generation: Creating functionally zoned and more plausible urban environments for simulations, games, and digital twins.
• Educational Tools: Demonstrating the interplay of transport infrastructure, accessibility, and land use zoning in urban systems.
• Research Platform: Serving as a foundation for extensions incorporating dynamic networks, multiple agent types, more sophisticated economic models, or alternative learning paradigms.

Conclusion

This research presents a valuable step towards integrating established urban theories (LUTI models) into procedural city generation using an agent-based reinforcement learning framework. By explicitly modeling accessibility derived from a road network graph and encoding zoning preferences into the agent's reward function, the system generates land use patterns that reflect fundamental urban spatial organization principles. While scalability and the static nature of the transport network remain key limitations, the methodology provides a novel approach for creating more functionally realistic artificial cities.